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Inequalities

# Inequalities - Solving Inequalities An inequality is the...

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From MathMotivation.com – Permission Granted For Use and Modification For Non-Profit Purposes Solving Inequalities An inequality is the result of replacing the = sign in an equation with <, >, , or . For example, 3x – 2 < 7 is a linear inequality. We call it “linear” because if the < were replaced with an = sign, it would be a linear equation. Inequalities involving polynomials of degree 2 or more, like 2x 3 – x > 0, are referred to as polynomial inequalities, or quadratic inequalities if the degree is exactly 2. Inequalities involving rational expressions are called rational inequalities. Some often used inequalities also involve absolute value expressions. Solving Inequalities: A Summary In a nutshell, solving inequalities is about one thing: sign changes. Find all the points at which there are sign changes - we call these points critical values . Then determine which, if any, of the intervals bounded by these critical values result in a solution. The solution to the inequality will consist of the set of all points contained by the solution intervals. Method To Solve Linear, Polynomial, or Absolute Value Inequalities: 1. Move all terms to one side of the inequality sign by applying the Addition, Subtraction, Multiplication, and Division Properties of Inequalities. You should have only zero on one side of the inequality sign. 2. Solve the associated equation using an appropriate method. This solution or solutions will make up the set of critical values. At these values, sign changes occur in the inequality. 3. Plot the critical values on a number line. Use closed circles for and inequalities, and use open circles ο for < and > inequalities. 4. Test each interval defined by the critical values. If an interval satisfies the inequality, then it is part of the solution. If it does not satisfy the inequality, then it is not part of the solution.

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